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Chapter 7 | Work, Energy, and Energy Resources
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(7.55)
(7.56)
so that
where is the total work done by all nonconservative forces and is the total work done by all conservative forces.
Figure 7.16 A person pushes a crate up a ramp, doing work on the crate. Friction and gravitational force (not shown) also do work on the crate; both forces oppose the person’s push. As the crate is pushed up the ramp, it gains mechanical energy, implying that the work done by the person is greater than the work done by friction.
Consider Figure 7.16, in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that .
Substituting this equation into the previous one and solving for gives
(7.57)
This equation means that the total mechanical energy changes by exactly the amount of work done by
nonconservative forces. In Figure 7.16, this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.
We rearrange to obtain
(7.58)
This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If is
positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in Figure 7.16. If is
negative, then mechanical energy is decreased, such as when the rock hits the ground in Figure 7.15(b). If is zero, then
mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy.
Applying Energy Conservation with Nonconservative Forces
When no change in potential energy occurs, applying amounts to applying the work-energy
theorem by setting the change in kinetic energy to be equal to the net work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that involve changes in both potential and kinetic energy, the previous equation
says that you can start by finding the change in mechanical energy that would have resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved.
Example 7.9 Calculating Distance Traveled: How Far a Baseball Player Slides
Consider the situation shown in Figure 7.17, where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N.
